In this discussion, we are going to discuss concepts of moments and centers of mass.
Practice problems for finding moments and centers of mass will be in another post.
This post is for those who wish to understand moments and centers of mass in depth, beginning with the basics.
Below is an outline of the different dimensions discussed regarding moments and centers of mass:
Along a Line
Beginning With the Basics
A mass is a quantity of matter that is obtained in an object and is measured by weight.
A moment (of a mass) is equal to the mass times the distance from the axis of rotation.
The mass of a system is the sum of all masses.
The moment of a system is the sum of all moments.
The center of mass is the sum of all moments divided by the sum of all masses.
For illustrations in this discussion, let the axis of rotation be the origin.
Moments and Centers of Mass: (Along a Line)
Given a seesaw, let the fulcrum is placed at the origin.
Supposed that:
mass1 is placed at some position x1
mass2 is placed at some position x2
mass3 is placed at some position x3.
Let
\widetilde{x_1}=Distance\,of\, mass\,from\,origin.
Then
m_{1} \,\,is\,\,\widetilde{x_1}\,\,units\, \,from\, \,the\,\, origin
m_{2} \,\,is\,\,\widetilde{x_2}\,\,units\, \,from\, \,the\,\, origin
m_{3} \,\,is\,\,\widetilde{x_3}\,\,units\, \,from\, \,the\,\, origin
Center of One Mass
Center of One Mass = Distance of Mass From the Origin
\widetilde{x}\,=\, x
The Moment of One Mass About the Origin
Moment of One Mass About the Origin = (Mass) x(The Distance From the Origin)
M_{0}\,=\,m\widetilde{x}
M_{0}\,=\,mx
The Mass of the System
Mass of system = The Sum of All Masses
M0 = m1 + m3 + m3
The Moment of the System
Moment of system = The Sum of All Moments
M_{0}\,=\,m_{1}\widetilde{x_1}\,+\,m_{2}\widetilde{x_2}\,+m_{3}\widetilde{x_3}
M_{0}\,=\,m_{1}x_{1}\,+\,m_{2}x_{2}\,+\,m_{3}x_{3}
The Center of Mass of the Seesaw
\bar{x}=\frac{Sum \, of \, all \, moments}{Sum \, of \, all\, masses}
\bar{x}=\frac{M_0}{m}
The system is balanced at the center of mass
\bar{x}
Go to Page 2 for: Moments and Centers of Mass Distributed Over a Plane Region